Wave nature of electron (English)

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4-4: The Wave Nature of Electrons

As learned

in Part 1 and 2,

it has been made clear

that,

if a substance

is divided into

finer and finer pieces,

we reach molecules

and atoms,

and then we realize

that the atoms consist of

electrons and nuclei.

Namely, it has been

clarified that

matter is a collection

of ultramicroscopic

particles.

In the classical physics

up to the 19th century,

these particles were

considered to move

obeying Newtonian mechanics

and Maxwellian electromagnetism.

However, this viewpoint

has become doubtful

after the proposal of the Bohr model

of the atomic structure

(Bohr's quantum theory).

On the other hand,

light

had been considered

to be electromagnetic waves

in the classical physics.

However, after

the discovery of light quanta (photons),

it was clarified

that the light has

wave nature at one time

and

particle nature

at another time.

Namely, light has

a kind of duality.

Thinking of this,

L. V. de Broglie

(France, 1892 - 1987)

guessed that such substance

particles as electrons or

protons might have

wave nature, although

they had been considered

to have particle nature

so far.

[De Broglie Waves]

The idea of

de Broglie waves

de Broglie matter waves

is that,

thinking of the fact

that light which had been

considered to be

waves had the particle nature,

such a substance particle

as electron or proton

which had been considered

to be "particle"

might have "wave nature".

Hence, de Broglie

thought that,

if the double aspect,

i.e.

the duality

of the wave nature

and the particle nature,

is valid not only

for light but also

for such substance

particles as electrons,

Einstein's relations

which connect the particle

and wave aspects

in light quanta

would be satisfied

for de Broglie matter

waves as well.

Therefore the relations,

Eq. (1),

are often called

Einstein - de Broglie's relations.

If we apply these relations

to the case of the Bohr model

of the hydrogen atom,

we can well understand

its plausibility

as follows.

If we consider

that the electron

in a hydrogen atom

moves at constant speed

along a circular orbit

around the nucleus

(proton), the quantum condition

in Bohr's quantum theory

is written as

Eq. (3) on the preceding page.

By using Einstein's relation

in this equation,

the quantum condition

is written

This equation means

that the circumference

of the circular orbit

of the electron

must be a integral multiple

of the wavelength of

de Broglie wave.

In other word,

de Broglie wave

accompanying the motion

of the electron

should be

continuous.

Therefore, we can easily

understand the quantum condition

that determines

the stationary states

by considering the continuity

of de Broglie waves.

(See the following figure.)

Bohr's quantum condition. The condition for stationary states

The circumference of

the circular orbit

of the electron

should be an integral multiple

of the wavelength

of de Broglie wave,

otherwise the wave

cannot be smoothly continuous.

[The Laue Diagram -- Wave Nature of X-Rays]

In 1914,

M. T. F. von Laue

(Germany, 1879 - 1960)

found a symmetrical pattern

of spots

on a photographic plate

by a beam of X-rays

that had passed

through a crystal.

This is called the

Laue diagram,

which is produced

by interference

of beams of X-rays diffracted

on the different atomic planes

in the crystal.

It was confirmed

by this phenomena that

X-rays are electromagnetic

waves of very short wavelengths.

The reason why

the interference pattern

in the Laue diagram

appears is as follows:

As shown in the above

figure,

an X-ray beam

is irradiated on a crystal

consisting of the atomic

planes A, B, ...

As light reflects

on a mirror,

so the X-ray beam striking

the plane A

with

the incident angle

reflects most strongly

in the same angle .

The similar happens

on the plane B.

These X-ray beams reflected

on the planes A and B

interfere with each other and

constructive interference

takes place when

the difference in pathlength

is equal to an integral

number of wavelength

i.e.,

where d is

the interplanar distance,

and

is the angle

between

the incident X-ray

and the crystal plane.

This is called

Bragg's condition.

Thus a bright spot

will appear

in the angle satisfying

the condition.

When the white X-rays

which contain all wavelengths

are illuminated on a crystal,

various atomic planes

in the crystal

reflect the X-rays satisfying

Bragg's condition

selectively or exclusively,

and then

a symmetric pattern

of spots is produced

on the photographic plate

placed behind.

This is the Laue diagram,

an example of which

is shown in the following

picture.

An example of the Laue diagram

This is a interference

pattern of X-rays

diffracted by

a single crystal of silicon.

The small black spots

lined crosswise

are the so-called Laue spots.

Here, the wavelength

and the interatomic distance

are about

(Courtesy of Prof. Y. Soejima,

Dept. of Physics,

Kyushu Univ.)

[The Empirical Evidences of the Wave Nature of Electrons]

The wavelength of

de Broglie wave

associated with the electrons

accelerated by

an electric potential

of 100 V

is about

().

This is almost

the same as that

of ordinary X-rays.

It is therefore supposed

that, if we bombarded

this electron beam

on a crystal,

we would observe

a diffraction pattern

similar to

the Laue diagram

in the case of X-rays.

In 1927,

American physicists,

C. J. Davisson

(USA, 1881 - 1958) and

L. H. Germer

(USA, 1896 - 1971)

observed an interference

pattern of

electron beams

diffracted by a single

crystal of nickel

for the first time

and, in the same year,

G. P. Thomson

(UK, 1892 - 1975)

independently observed

the diffraction pattern

using a metallic multi-crystal.

Next year,

S. Kikuchi

(Japan, 1902 - 74)

was also successful

in a similar experiment

using a thin film of mica.

An example of the diffraction

and interference

phenomena of electrons

is shown in the following

picture.

A diffraction pattern of electron beams

Electron beams

are diffracted

by a crystal

of manganese-nickel alloy.

In this case, the de Broglie wavelength

is shorter than

which corresponds

to rather high speed

electron beam.

(Courtesy of Prof. Y. Soejima,

Dept. of Physics,

Kyushu Univ.)

Looking at these results,

people could no longer

deny that

electrons possess

the double nature

(dual property),

i.e. the particle nature

and the wave nature.

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